Space Time and Gravitation/Chapter 7

by Arthur Eddington

 

CHAPTER VII

WEIGHING LIGHT

Query 1. Do not Bodies act upon Light at a distance, and by their action bend its Rays, and is not this action (caeteris paribus) strongest at the least distance?

Newton, Opticks.

We come now to the experimental test of the influence of gravitation on light discussed theoretically in the last chapter. It is not the general purpose of this book to enter into details of experiments; and if we followed this plan consistently, we should, as hitherto, summarise the results of the observations in a few lines. But it is this particular test which has turned public attention towards the relativity theory, and there appears to be widespread desire for information. We shall therefore tell the story of the eclipse expeditions in some detail. It will make a break in the long theoretical arguments, and will illustrate the important applications of this theory to practical observations.

It must be understood that there were two questions to answer: firstly, whether light has weight (as suggested by Newton), or is indifferent to gravitation; secondly, if it has weight, is the amount of the deflection in accordance with Einstein's or Newton's laws?

It was already known that light possesses mass or inertia like other forms of electromagnetic energy. This is manifested in the phenomena of radiation-pressure. Some force is required to stop a beam of light by holding an obstacle in its path; a searchlight experiences a minute force of recoil just as if it were a machine-gun firing material projectiles. The force, which is predicted by orthodox electromagnetic theory, is exceedingly minute; but delicate experiments have detected it. Probably this inertia of radiation is of great cosmical importance, playing a great part in the equilibrium of the more diffuse stars. Indeed it is probably the agent which has carved the material of the universe into stars of roughly uniform mass. Possibly the tails of comets are a witness to the power of the momentum of sunlight, which drives outwards the smaller or the more absorptive particles.

It is legitimate to speak of a pound of light as we speak of a pound of any other substance. The mass of ordinary quantities of light is however extremely small, and I have calculated that at the low charge of 3d. a unit, an Electric Light Company would have to sell light at the rate of £140,000,000 a pound. All the sunlight falling on the earth amounts to 160 tons daily.

It is perhaps not easy to realise how a wave-motion can have inertia, and it is still more difficult to understand what is meant by its having weight. Perhaps this will be better understood if we put the problem in a concrete form. Imagine a hollow body, with radiant heat or light-waves traversing the hollow; the mass of the body will be the sum of the masses of the material and of the radiant energy in the hollow; a greater force will be required to shift it because of the light-waves contained in it. Now let us weigh it with scales or a spring-balance. Will it also weigh heavier on account of the radiation contained, or will the weight be that of the solid material alone? If the former, then clearly from this aspect light has weight; and it is not difficult to deduce the effect of this weight on a freely moving light-beam not enclosed within a hollow.

The effect of weight is that the radiation in the hollow body acquires each second a downward momentum proportional to its mass. This in the long run is transmitted to the material enclosing it. For a free light-wave in space, the added momentum combines with the original momentum, and the total momentum determines the direction of the ray, which is accordingly bent. Newton's theory suggests no means for bringing about the bending, but contents itself with predicting it on general principles. Einstein's theory provides a means, viz. the variation of velocity of the waves.

Hitherto mass and weight have always been found associated in strict proportionality. One very important test had already shown that this proportionality is not confined to material energy. The substance uranium contains a great deal of radio-active energy, presumably of an electromagnetic nature, which it slowly liberates. The mass of this energy must be an appreciable fraction of the whole mass of the substance. But it was shown by experiments with the Eötvös torsion-balance that the ratio of weight to mass for uranium is the same as for all other substances; so the energy of radio-activity has weight. Still even this experiment deals only with bound electromagnetic energy, and we are not justified in deducing the properties of the free energy of light.

It is easy to see that a terrestrial experiment has at present no chance of success. If the mass and weight of light are in the same proportion as for matter, the ray of light will be bent just like the trajectory of a material particle. On the earth a rifle bullet, like everything else, drops 16 feet in the first second, 64 feet in two seconds, and so on, below its original line of flight; the rifle must thus be aimed above the target. Light would also drop 16 feet in the first second^[1]; but, since it has travelled 186,000 miles along its course in that time, the bend is inappreciable.

In fact any terrestrial course is described so quickly that gravitation has scarcely had time to accomplish anything.

The experiment is therefore transferred to the neighbourhood of the sun. There we get a pull of gravitation 27 times more intense than on the earth; and—what is more important—the greater size of the sun permits a much longer trajectory throughout which the gravitation is reasonably powerful. The deflection in this case may amount to something of the order of a second of arc, which for the astronomer is a fairly large quantity.

In Fig. 16 the line $EFQP$ shows the track of a ray of light from a distant star $P$ which reaches the earth $E$. The main part of the bending of the ray occurs as it passes the sun $S$; and the initial course $PQ$ and the final course $FE$ are practically straight. Since the light rays enter the observer's eye or telescope in the direction $FE$, this will be the direction in which the star appears. But its true direction from the earth is $QP$, the initial course. So the star appears displaced outwards from its true position by an angle equal to the total deflection of the light.

It must be noticed that this is only true because a star is so remote that its true direction with respect to the earth $E$ is indistinguishable from its direction with respect to the point $Q$. For a source of light within the solar system, the apparent displacement of the source is by no means equal to the deflection of the light-ray. It is perhaps curious that the attraction of light by the sun should produce an apparent displacement of the star away from the sun; but the necessity for this is clear.

The bending affects stars seen near the sun, and accordingly the only chance of making the observation is during a total eclipse when the moon cuts off the dazzling light. Even then there is a great deal of light from the sun's corona which stretches far above the disc. It is thus necessary to have rather bright stars near the sun, which will not be lost in the glare of the corona. Further the displacements of these stars can only be measured relatively to other stars, preferably more distant from the sun and less displaced; we need therefore a reasonable number of outer bright stars to serve as reference points.

In a superstitious age a natural philosopher wishing to perform an important experiment would consult an astrologer to ascertain an auspicious moment for the trial. With better reason, an astronomer to-day consulting the stars would announce that the most favourable day of the year for weighing light is May 29. The reason is that the sun in its annual journey round the ecliptic goes through fields of stars of varying richness, but on May 29 it is in the midst of a quite exceptional patch of bright stars—part of the Hyades—by far the best star-field encountered. Now if this problem had been put forward at some other period of history, it might have been necessary to wait some thousands of years for a total eclipse of the sun to happen on the lucky date. But by strange good fortune an eclipse did happen on May 29, 1919. Owing to the curious sequence of eclipses a similar opportunity will recur in 1938; we are in the midst of the most favourable cycle. It is not suggested that it is impossible to make the test at other eclipses; but the work will necessarily be more difficult.

Attention was called to this remarkable opportunity by the Astronomer Royal in March, 1917; and preparations were begun by a Committee of the Royal Society and Royal Astronomical Society for making the observations. Two expeditions were sent to different places on the line of totality to minimise the risk of failure by bad weather. Dr A. C. D. Crommelin and Mr C. Davidson went to Sobral in North Brazil; Mr E. T. Cottingham and the writer went to the Isle of Principe in the Gulf of Guinea, West Africa. The instrumental equipment for both expeditions was prepared at Greenwich Observatory under the care of the Astronomer Royal; and here Mr Davidson made the arrangements which were the main factor in the success of both parties.

The circumstances of the two expeditions were somewhat different and it is scarcely possible to treat them together. We shall at first follow the fortunes of the Principe observers. They had a telescope of focal length 11 feet 4 inches. On their photographs 1 second of arc (which was about the largest displacement to be measured) corresponds to about  1/1500 inch—by no means an inappreciable quantity. The aperture of the object-glass was 13 inches, but as used it was stopped down to 8 inches to give sharper images. It is necessary, even when the exposure is only a few seconds, to allow for the diurnal motion of the stars across the sky, making the telescope move so as to follow them. But since it is difficult to mount a long and heavy telescope in the necessary manner in a temporary installation in a remote part of the globe, the usual practice at eclipses is to keep the telescope rigid and reflect the stars into it by a coelostat—a plane mirror kept revolving at the right rate by clock-work. This arrangement was adopted by both expeditions.

The observers had rather more than a month on the island to make their preparations. On the day of the eclipse the weather was unfavourable. When totality began the dark disc of the moon surrounded by the corona was visible through cloud, much as the moon often appears through cloud on a night when no stars can be seen. There was nothing for it but to carry out the arranged programme and hope for the best. One observer was kept occupied changing the plates in rapid succession, whilst the other gave the exposures of the required length with a screen held in front of the object-glass to avoid shaking the telescope in any way.

For in and out, above, about, below
'Tis nothing but a Magic Shadow-show
Played in a Box whose candle is the Sun
Round which we Phantom Figures come and go.

Our shadow-box takes up all our attention. There is a marvellous spectacle above, and, as the photographs afterwards revealed, a wonderful prominence-flame is poised a hundred thousand miles above the surface of the sun. We have no time to snatch a glance at it. We are conscious only of the weird half-light of the landscape and the hush of nature, broken by the calls of the observers, and beat of the metronome ticking out the 302 seconds of totality.

Sixteen photographs were obtained, with exposures ranging from 2 to 20 seconds. The earlier photographs showed no stars, though they portrayed the remarkable prominence; but apparently the cloud lightened somewhat towards the end of totality, and a few images appeared on the later plates. In many cases one or other of the most essential stars was missing through cloud, and no use could be made of them; but one plate was found showing fairly good images of five stars, which were suitable for a determination. This was measured on the spot a few days after the eclipse in a micrometric measuring-machine. The problem was to determine how the apparent positions of the stars, affected by the sun's gravitational field, compared with the normal positions on a photograph taken when the sun was out of the way. Normal photographs for comparison had been taken with the same telescope in England in January. The eclipse photograph and a comparison photograph were placed film to film in the measuring-machine so that corresponding images fell close together^[2], and the small distances were measured in two rectangular directions. From these the relative displacements of the stars could be ascertained. In comparing two plates, various allowances have to be made for refraction, aberration, plate-orientation, etc.; but since these occur equally in determinations of stellar parallax, for which much greater accuracy is required, the necessary procedure is well-known to astronomers.

The results from this plate gave a definite displacement, in good accordance with Einstein's theory and disagreeing with the Newtonian prediction. Although the material was very meagre compared with what had been hoped for, the writer (who it must be admitted was not altogether unbiassed) believed it convincing.

It was not until after the return to England that any further confirmation was forthcoming. Four plates were brought home undeveloped, as they were of a brand which would not stand development in the hot climate. One of these was found to show sufficient stars; and on measurement it also showed the deflection predicted by Einstein, confirming the other plate.

The bugbear of possible systematic error affects all investigations of this kind. How do you know that there is not something in your apparatus responsible for this apparent deflection? Your object-glass has been shaken up by travelling; you have introduced a mirror into your optical system; perhaps the 50° rise of temperature between the climate at the equator and England in winter has done some kind of mischief. To meet this criticism, a different field of stars was photographed at night in Principe and also in England at the same altitude as the eclipse field. If the deflection were really instrumental, stars on these plates should show relative displacements of a similar kind to those on the eclipse plates. But on measuring these check-plates no appreciable displacements were found. That seems to be satisfactory evidence that the displacement observed during the eclipse is really due to the sun being in the region, and is not due to differences in instrumental conditions between England and Principe. Indeed the only possible loophole is a difference between the night conditions at Principe when the check-plates were taken, and the day, or rather eclipse, conditions when the eclipse photographs were taken. That seems impossible since the temperature at Principe did not vary more than 1° between day and night.

The problem appeared to be settled almost beyond doubt; and it was with some confidence that we awaited the return of the other expedition from Brazil. The Brazil party had had fine weather and had gained far more extensive material on their plates. They had remained two months after the eclipse to photograph the same region before dawn, when clear of the sun, in order that they might have comparison photographs taken under exactly the same circumstances. One set of photographs was secured with a telescope similar to that used at Principe. In addition they used a longer telescope of 4 inches aperture and 19 feet focal length^[3]. The photographs obtained with the former were disappointing. Although the full number of stars expected (about 12) were shown, and numerous plates had been obtained, the definition of the images had been spoiled by some cause, probably distortion of the coelostat-mirror by the heat of the sunshine falling on it. The observers were pessimistic as to the value of these photographs; but they were the first to be measured on return to England, and the results came as a great surprise after the indications of the Principe plates. The measures pointed with all too good agreement to the "half-deflection" that is to say, the Newtonian value which is one-half the amount required by Einstein's theory. It seemed difficult to pit the meagre material of Principe against the wealth of data secured from the clear sky of Sobral. It is true the Sobral images were condemned, but whether so far as to invalidate their testimony on this point was not at first clear; besides the Principe images were not particularly well-defined, and were much enfeebled by cloud. Certain compensating advantages of the latter were better appreciated later. Their strong point was the satisfactory check against systematic error afforded by the photographs of the check-field; there were no check-plates taken at Sobral, and, since it was obvious that the discordance of the two results depended on systematic error and not on the wealth of material, this distinctly favoured the Principe results. Further, at Principe there could be no evil effects from the sun's rays on the mirror, for the sun had withdrawn all too shyly behind the veil of cloud. A further advantage was provided by the check-plates at Principe, which gave an independent determination of the difference of scale of the telescope as used in England and at the eclipse; for the Sobral plates this scale-difference was eliminated by the method of reduction, with the consequence that the results depended on the measurement of a much smaller relative displacement.

There remained a set of seven plates taken at Sobral with the 4-inch lens; their measurement had been delayed by the necessity of modifying a micrometer to hold them, since they were of unusual size. From the first no one entertained any doubt that the final decision must rest with them, since the images were almost ideal, and they were on a larger scale than the other photographs. The use of this instrument must have presented considerable difficulties—the unwieldy length of the telescope, the slower speed of the lens necessitating longer exposures and more accurate driving of the clock-work, the larger scale rendering the focus more sensitive to disturbances—but the observers achieved success, and the perfection of the negatives surpassed anything that could have been hoped for.

These plates were now measured and they gave a final verdict definitely confirming Einstein's value of the deflection, in agreement with the results obtained at Principe.

It will be remembered that Einstein's theory predicts a deflection of 1″.74 at the edge of the sun^[4], the amount falling off inversely as the distance from the sun's centre. The simple Newtonian deflection is half this, 0″.87. The final results (reduced to the edge of the sun) obtained at Sobral and Principe with their "probable accidental errors" were

Sobral	1″.98 ± 0″.12,
Principe	1″.61 ± 0″.80.

It is usual to allow a margin of safety of about twice the probable error on either side of the mean. The evidence of the Principe plates is thus just about sufficient to rule out the possibility of the "half-deflection," and the Sobral plates exclude it with practical certainty. The value of the material found at Principe cannot be put higher than about one-sixth of that at Sobral; but it certainly makes it less easy to bring criticism against this confirmation of Einstein's theory seeing that it was obtained independently with two different instruments at different places and with different kinds of checks.

The best check on the results obtained with the 4-inch lens at Sobral is the striking internal accordance of the measures for different stars. The theoretical deflection should vary inversely as the distance from the sun's centre; hence, if we plot the mean radial displacement found for each star separately against the inverse distance, the points should lie on a straight line. This

is shown in Fig. 17 where the broken line shows the theoretical prediction of Einstein, the deviations being within the accidental errors of the determinations. A line of half the slope representing the half-deflection would clearly be inadmissible.

Moreover, values of the deflection were deduced from the measures in right ascension and declination independently. These were in close agreement.

A diagram showing the relative positions of the stars is given in Fig. 18.

The square shows the limits of the plates used at Principe, and the oblique rectangle the limits with the 4-inch lens at Sobral. The centre of the sun moved from $S$ to $P$ in the 2 1/4

hours interval between totality at the two stations; the sun is here represented for a time about midway between. The stars measured on the Principe plates were Nos. 3, 4, 5, 6, 10, 11; those at Sobral were 11, 10, 6, 5, 4, 2, 3 (in the order of the dots from left to right in Fig. 17). None of these were fainter than 6^m.0, the brightest κ¹ Tauri (No. 4) being 4^m.5.

It has been objected that although the observations establish a deflection of light in passing the sun equal to that predicted by Einstein, it is not immediately obvious that this deflection must necessarily be attributed to the sun's gravitational field. It is suggested that it may not be an essential effect of the sun as a massive body, but an accidental effect owing to the circumstance that the sun is surrounded by a corona which acts as a refracting atmosphere. It would be a strange coincidence if this atmosphere imitated the theoretical law in the exact quantitative way shown in Fig. 17; and the suggestion appears to us far-fetched. However the objection can be met in a more direct way. We have already shown that the gravitational effect on light is equivalent to that produced by a refracting medium round the sun and have calculated the necessary refractive index. At a height of 400,000 miles above the surface the refractive index required is 1.0000021. This corresponds to air at  1/140 atmosphere, hydrogen at  1/70 atmosphere, helium at  1/20 atmospheric pressure. It seems obvious that there can be no material of this order of density at such a distance from the sun. The pressure on the sun's surface of the columns of material involved would be of the order 10,000 atmospheres; and we know from spectroscopic evidence that there is no pressure of this order. If it is urged that the mass could perhaps be supported by electrical forces, the argument from absorption is even more cogent. The light from the stars photographed during the eclipse has passed through a depth of at least a million miles of material of this order of density—or say the equivalent of 10,000 miles of air at atmospheric density. We know to our cost what absorption the earth's 5 miles of homogeneous atmosphere can effect. And yet at the eclipse the stars appeared on the photographs with their normal brightness. If the irrepressible critic insists that the material round the sun may be composed of some new element with properties unlike any material known to us, we may reply that the mechanism of refraction and of absorption is the same, and there is a limit to the possibility of refraction without appreciable absorption. Finally it would be necessary to arrange that the density of the material falls off inversely as the distance from the sun's centre in order to give the required variation of refractive index.

Several comets have been known to approach the sun within the limits of distance here considered. If they had to pass through an atmosphere of the density required to account for the displacement, they would have suffered enormous resistance. Dr Crommelin has shown that a study of these comets sets an upper limit to the density of the corona, which makes the refractive effect quite negligible.

Those who regard Einstein's law of gravitation as a natural deduction from a theory based on the minimum of hypotheses will be satisfied to find that his remarkable prediction is quantitatively confirmed by observation, and that no unforeseen cause has appeared to invalidate the test.

1. Or 32 feet according to Einstein's law. The fall increases with the speed of the motion.
2. This was possible because at Principe the field of stars was reflected in the coelostat mirror, whereas in England it was photographed direct.
3. See Frontispiece. The two telescopes are shown and the backs of the two coelostat-mirrors which reflect the sky into them. The clock driving the larger mirror is seen on the pedestal on the left.
4. The predicted deflection of light from infinity to infinity is just over 1″.745, from infinity to the earth it is just under.